Park City Mathematics Institute
Secondary School Teachers Program
Summer 2011
Week 2: Monday Zome Building

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Reasoning from Data and Chance
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Ning: PCMI 2011

Forms

The large Zometool construction for 2011 is a compound of a 120-cell and a Gosset polytope projected into three dimensions. The 120-cell is a four dimensional analog of a dodecahedron, a longtime PCMI favorite [http://mathforum.org/pcmi/truncated120.html]. The Gosset polytope is an eight dimension structure important in geometry and algebra [http://vzome.com/gosset/].

Why combine these two things, which don't even live in the same dimension? You may know that the dodecahedron and icosahedron are duals: put a vertex in each face of one polyhedron, connect adjacent vertices, and you get the other. A Zome creation showing concentric dodecahedron and icosahedron of roughly the same diameter is an inspiration for what we made.

The 120-cell is dual to a four dimensional polytope known as the 600 cell. A projection of the 600 cell into three dimensions can be built in the Zome system, but the model is dense to the point of obscuring structure. The Gosset polytope, however, can be built from two copies of the 600-cell and has a "nicer" projection into three dimensions. The large compound shape shows traits of both structures, e.g., the blue tunnels of the 120-cell and the parallel planes in the Gosset polytope, while illustrating duality. In fact, the compound dodecahedron and icosahedron structure appears three times in layers of our primary 2011 giant Zome build, created under the guidance and enthusiasm of ZomeTool "chief visionary officer" Paul Hildebrandt.

--Brian Hopkins

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© 2001 - 2014 Park City Mathematics Institute
IAS/Park City Mathematics Institute is an outreach program of the Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540.
Send questions or comments to: Suzanne Alejandre and Jim King

With program support provided by Math for America

This material is based upon work supported by the National Science Foundation under Grant No. 0314808 and Grant No. ESI-0554309. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.